Rank Distributions for Independent Normals with a Single Outlier

Abstract

Thurstone's latent-normal model, introduced a century ago to describe human preferences in psychometrics (1927), remains a cornerstone for modeling random rankings. Yet when the underlying normals differ in distribution, the joint law of ranks $R_{i}:=\sum_{j=1}^{n}\mathbf{1}_{X_{j}\leq X_{i}}$ is virtually unexplored. We study the simplest non-identically-distributed case: $n+1$ independent normals with $X_{0}\sim\mathcal{N}\left(\mu_{0},\,\sigma_{0}^{2}\right)$ and $X_{i}\sim\mathcal{N}\left(\mu,\,\sigma^{2}\right)$ for $1\leq i\leq n$. Here, $R_0 \mid X_0 \;\sim\; 1 + \mathrm{Binomial}\bigl(n,\;\Phi\bigl(\bigl(X_0 - \mu\bigr)\big/\sigma\bigr)\bigr)$, and the success probability $\Phi\bigl(\bigl(X_0 - \mu\bigr)\big/\sigma\bigr)$ is accurately modeled by a beta distribution. Exploiting beta-binomial conjugacy, we observe that $R_{0}-1$ follows a beta-binomial law, which then yields a precise approximation for the joint distribution of $\left(R_{0},R_{i_{1}},\ldots,R_{i_{m}}\right)$. We derive closed-form expressions for $\mathbb{E}R_{i}$, $\mathrm{Cov}\left(R_{i},R_{j}\right)$, and the limiting distributions of $\left(R_{0},R_{i_{1}},\ldots,R_{i_{m}}\right)$ as key parameters grow large or small.